Question

For any ring k, let A subspace of M(2,k) where a11+a21=a12+a22. Show that A is a subring of M2(k), and that it is isomorphic to the ring R of 2 × 2 lower triangular matrices over k.

Answer 1

Let $\alpha = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ . Then $\alpha^{-1}R\alpha$ consists of the matrices

$$\left( \begin{array}{cc} 1 & -1 \\ 0 & 1 \end{array} \right) \left( \begin{array}{cc} x & 0 \\ y & z \end{array} \right) \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) = \left( \begin{array}{cc} x - y & x - y - z \\ y & y + z \end{array} \right)$$

We see easily that the set of these matrices is exactly $A$ . Therefore, $A$ is just a "conjugate" of the subring $R$ in the ring $\mathbf{M}2(k)$ . In particular, $A \sim R$ .